Download CONDENSED MATTER: towards Absolute Zero CONDENSED

CONDENSED MATTER:
towards Absolute Zero
The lowest temperatures
reached for bulk matter
between 1970- 2000 AD.
PCES 16.8
We have seen the voyages to
inner & outer space in physics.
There is also a voyage to the
ultra-cold, which is now within
10-10 K of absolute zero (see left).
One can never reach absolute
zero (at which all random
thermal motion stops), since no
cooling device is perfectly
reversible- something always
leaks back.
The fascination of ultralow T is
that more & more complex
kinds of ‘quantum order’
develop, undisturbed by the
thermal motion. This has led to
some of the most extraordinary
phenomena in physics.
The experimental techniques
which get to such temperatures
are equally remarkable.
CONDENSED MATTER: Superfluidity
The ‘ROTA 2’ rotating cryostat.
It cools to roughly 0.5 mK. The
entire 500 kg apparatus can turn
up to 6 times per second
PCES 16.9
Superfluidity is defined by the absence of viscous resistance
to the flow of a fluid. The superfluid flows freely, ad infinitum,
through holes hardly larger than atomic size. The ‘fountain
effect’ at right shows free flow of He-4 liquid through packed
‘jeweller’s rouge’ (rather like lipstick).
The ultimate explanation of this is in the Bose statistics of the
particles. He-4 atoms are bosons (with 2 electrons, 2 protons, &
2 neutrons). At low T they all ‘Bose condense’ into the same
quantum state. The superfluidity then arises because it takes a
Fountain effect
finite energy to excite the system out of this state- only possible
if it flows faster than a ‘critical
velocity’ vc, or if some object
moves through it faster than
than vc .
He-3 (1 neutron
instead of 2) is a fermion- but
forms ‘Cooper pairs’ of atoms,
which behave as bosons.‘Pairbreaking’ again occurs above
a critical velocity (left).
Wire in He-3 superfluid moves with zero
resistance until a critical velocity, then
emits ‘wind’ of ‘broken pair’excitations
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MACROSCOPIC WAVE-FUNCTIONS
If all the particles in the system end up in the same
state, we can write down a ‘macroscopic wave-function’
for the quantum state of the whole system! It was first
seen by London in 1938 that this was the key to
superfluidity & superconductivity. Later Landau gave
a more complete theory (1941), and then finally in 1957
F London
the BCS (Bardeen-Cooper-Schrieffer) theory, & an
equivalent theory of Bogoliubov, gave a definitive explanation of these
phenomena. The macroscopic quantum state is written in the form
Ψ (r1, r2, …rN)
Bogoliubov
J Bardeen
PCES 16.10
LD Landau
(1908-1968)
= Σperm φ (r1) φ (r2) …φ (rN)
where the sum is over all possible swaps of the particles (remember
the particles are indistinguishable). This formula may look terrible,
but it just says that all particles are in the same quantum state φ.
Because all particles are in the same state,
we can just talk about a single quantum state
Ψ ( r ) for the whole superfluid. This is the
famous ‘macroscopic wave-function’ of
London’s. But it is still a probability
amplitude! London’s idea was disbelieved
LN Cooper
JR Schrieffer
when proposed, but is now a central part of
physics.
CONDENSED MATTER:
Quantum vortices in
Superfluids
PCES 16.11
Suppose we look at a vortex in a
superfluid- ie., fluid circulating around
a core. From what we saw with atoms
this tells us that we have probability
waves circulating round the core with
wavelength λ = h/p = h/mv, where v is
the velocity of the atoms circulating
around the core. But then we have the
same situation as with the atom- only
certain velocities are allowed, if we are
Different vortex patterns in superfluid He-3
to fit the waves around the core. Hence
we find that the total circulation is quantized- we have ‘quantized vortices’. In
this simple picture the core is like a string- in fact it has a finite
diameter. In He-4 this is very small (only about 1 Angstrom!), but
in other superfluids like He-3 it is much larger (~ 150 Angstroms),
& so the core is itself very complex (see above).
The vortices themselves are quantum excitations- so they also
have a probability density! They have fascinating propertiesCirculating
one of the most interesting’ is that they can form closed
superfluid around core
‘vortex rings’, which are also probability waves.
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CONDENSED MATTER:
Quantum Vortices in
Superconductors
Superconductivity is a condensation of pairs of
electrons, all into a single state. If we try to disturb
this macroscopic quantum state by applying an
external magnetic field, the supercurrents in the
system, flowing without resistance, simply adjust to
block the field from entering the superconductor
(the ‘Meissner effect’). In some materials the field
can get in via vortices, like those in superfluidsagain, the circulating current is quantised.
If we have a loop of
superconducting material
we can trap magnetic flux
inside it- this is kept out
of the superconductor by
currents in it, as before.
Again, the circulating
current is quantised, and
thus so is the flux- in units
of a flux quantum h/e
Magnetic Field through
superconducting ring
PCES 16.12
TOP: magnetic field lines around
a superconductor
MIDDLE: vortices penetrate
BOTTOM: close-up of vortex in
superconductor
PCES 16.13
NEUTRON STARS:
Stellar superfluids
Actually most of the matter in the universe
is in superfluid form. Neutron stars, left after
a supernova explosion, are actually like giant
nuclei, and they are superfluid. The neutron
star & the superfluid in it are rapidly rotating,
hence full of vortex lines. This is also a
circulating electric current (the protons are
charged), so a huge magnetic field is created.
LEFT: structure of a
neutron star. The inner
dense parts (containing
almost all mass) are
neutron & proton
superfluids
RIGHT: magnetic field
around neutron starhigh energy particles
are ejected along
magnetic poles
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The ‘SQUID’
PCES 16.14
As previously discussed (p. 12.6)
we can set up a 2-slit experiment in
superconductors using a “SQUID”
(Superconducting QUantum
Interference Device) ring. It
depends crucially on the existence
SQUID potential
2-slit interference in SQUID
of 2 ‘Josephson junctions’ which
allow flux to move in and out of the ring. The SQUID is a fantastically sensitive
detector of magnetic field- its interference pattern changes completely if a
single quantum of flux moves in or out of
the ring (assuming the electron waves are
coherent around the ring).
One can also show how the energy V of
the SQUID depends on the total magnetic
flux Φ through it (top right).
Moving a flux quantum in or
out means pushing the system
between 2 adjacent potential
wells. The current circulating
in the SQUID changes by a
large amount when this
transition occurs.
BD Josephson
Network of Josephson junctions and SQUIDs
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